We develop a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for the efficient numerical simulation of the parametric Navier–Stokes equations in the stream function-vorticity formulation. Unlike previous works, we choose different reduced coefficients for the vorticity and stream function fields. In addition, for parametric studies we use a global POD basis space obtained from a database of time dependent full order snapshots related to sample points in the parameter space. We test the performance of our ROM strategy with the well-known vortex merger benchmark and a more complex case study featuring the geometry of the North Atlantic Ocean. Accuracy and efficiency are assessed for both time reconstruction and physical parametrization.

%P 105536 %8 2022/06/14/ %@ 0045-7930 %G eng %U https://www.sciencedirect.com/science/article/pii/S0045793022001645 %! Computers & Fluids %0 Journal Article %J International Journal for Numerical Methods in Engineering %D 2022 %T Vibration Analysis of Piezoelectric Kirchhoff-Love shells based on Catmull-Clark Subdivision Surfaces %A Zhaowei Liu %A Andrew McBride %A Prashant Saxena %A Luca Heltai %A Yilin Qu %A Paul Steinmann %B International Journal for Numerical Methods in Engineering %G eng %0 Journal Article %J International Journal for Numerical Methods in Engineering %D 2021 %T A numerical approach for heat flux estimation in thin slabs continuous casting molds using data assimilation %A Umberto Emil Morelli %A Patricia Barral %A Peregrina Quintela %A Gianluigi Rozza %A Giovanni Stabile %B International Journal for Numerical Methods in Engineering %I Wiley %V 122 %P 4541–4574 %G eng %U https://doi.org/10.1002/nme.6713 %R 10.1002/nme.6713 %0 Journal Article %J Journal of Computational Physics %D 2021 %T A POD-Galerkin reduced order model for a LES filtering approach %A Michele Girfoglio %A Annalisa Quaini %A Gianluigi Rozza %XWe propose a Proper Orthogonal Decomposition (POD)-Galerkin based Reduced Order Model (ROM) for an implementation of the Leray model that combines a two-step algorithm called Evolve-Filter (EF) with a computationally efficient finite volume method. The main novelty of the proposed approach relies in applying spatial filtering both for the collection of the snapshots and in the reduced order model, as well as in considering the pressure field at reduced level. In both steps of the EF algorithm, velocity and pressure fields are approximated by using different POD basis and coefficients. For the reconstruction of the pressures fields, we use a pressure Poisson equation approach. We test our ROM on two benchmark problems: 2D and 3D unsteady flow past a cylinder at Reynolds number 0≤Re≤100. The accuracy of the reduced order model is assessed against results obtained with the full order model. For the 2D case, a parametric study with respect to the filtering radius is also presented.

%B Journal of Computational Physics %V 436 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85102138957&doi=10.1016%2fj.jcp.2021.110260&partnerID=40&md5=73115708267e80754f343561c26f4744 %R 10.1016/j.jcp.2021.110260 %0 Journal Article %J International Journal of Computational Fluid Dynamics %D 2020 %T Reduced Basis Model Order Reduction for Navier-Stokes equations in domains with walls of varying curvature %A Martin W. Hess %A Annalisa Quaini %A Gianluigi Rozza %XWe consider the Navier-Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.

%B International Journal of Computational Fluid Dynamics %V 34 %P 119-126 %G eng %U https://arxiv.org/abs/1901.03708 %R 10.1080/10618562.2019.1645328 %0 Journal Article %J International Journal of Computational Fluid Dynamics %D 2020 %T Reduced basis model order reduction for Navier–Stokes equations in domains with walls of varying curvature %A Martin W. Hess %A Annalisa Quaini %A Gianluigi Rozza %XWe consider the Navier–Stokes equations in a channel with a narrowing and walls of varying curvature. By applying the empirical interpolation method to generate an affine parameter dependency, the offline-online procedure can be used to compute reduced order solutions for parameter variations. The reduced-order space is computed from the steady-state snapshot solutions by a standard POD procedure. The model is discretised with high-order spectral element ansatz functions, resulting in 4752 degrees of freedom. The proposed reduced-order model produces accurate approximations of steady-state solutions for a wide range of geometries and kinematic viscosity values. The application that motivated the present study is the onset of asymmetries (i.e. symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the valve shape. Through our computational study, we found that the critical Reynolds number for the symmetry breaking increases as the wall curvature increases.

%B International Journal of Computational Fluid Dynamics %V 34 %P 119-126 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85085233294&doi=10.1080%2f10618562.2019.1645328&partnerID=40&md5=e2ed8f24c66376cdc8b5485aa400efb0 %R 10.1080/10618562.2019.1645328 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2020 %T A reduced order modeling technique to study bifurcating phenomena: Application to the gross-pitaevskii equation %A Federico Pichi %A Annalisa Quaini %A Gianluigi Rozza %XWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a reduced order modeling (ROM) technique, suitably supplemented with a hyperreduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called the Gross{Pitaevskii equation, as one or two physical parameters are varied. In the two-parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard full order method.

%B SIAM Journal on Scientific Computing %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85096768803&doi=10.1137%2f20M1313106&partnerID=40&md5=47d6012d10854c2f9a04b9737f870592 %R 10.1137/20M1313106 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2020 %T A Reduced Order technique to study bifurcating phenomena: application to the Gross-Pitaevskii equation %A Federico Pichi %A Annalisa Quaini %A Gianluigi Rozza %XWe propose a computationally efficient framework to treat nonlinear partial differential equations having bifurcating solutions as one or more physical control parameters are varied. Our focus is on steady bifurcations. Plotting a bifurcation diagram entails computing multiple solutions of a parametrized, nonlinear problem, which can be extremely expensive in terms of computational time. In order to reduce these demanding computational costs, our approach combines a continuation technique and Newton's method with a Reduced Order Modeling (ROM) technique, suitably supplemented with a hyper-reduction method. To demonstrate the effectiveness of our ROM approach, we trace the steady solution branches of a nonlinear Schrödinger equation, called Gross-Pitaevskii equation, as one or two physical parameters are varied. In the two parameter study, we show that our approach is 60 times faster in constructing a bifurcation diagram than a standard Full Order Method.

%B SIAM Journal on Scientific Computing %G eng %U https://arxiv.org/abs/1907.07082 %R https://doi.org/10.1137/20M1313106 %0 Journal Article %J Lecture Notes in Computational Science and Engineering %D 2020 %T A spectral element reduced basis method for navier–stokes equations with geometric variations %A Martin W. Hess %A Annalisa Quaini %A Gianluigi Rozza %XWe consider the Navier-Stokes equations in a channel with a narrowing of varying height. The model is discretized with high-order spectral element ansatz functions, resulting in 6372 degrees of freedom. The steady-state snapshot solutions define a reduced order space through a standard POD procedure. The reduced order space allows to accurately and efficiently evaluate the steady-state solutions for different geometries. In particular, we detail different aspects of implementing the reduced order model in combination with a spectral element discretization. It is shown that an expansion in element-wise local degrees of freedom can be combined with a reduced order modelling approach to enhance computational times in parametric many-query scenarios.

%B Lecture Notes in Computational Science and Engineering %V 134 %P 561-571 %G eng %R 10.1007/978-3-030-39647-3_45 %0 Journal Article %J Computers & Fluids %D 2019 %T A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization %A Michele Girfoglio %A Annalisa Quaini %A Gianluigi Rozza %XWe consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

%B Computers & Fluids %V 187 %P 27-45 %G eng %U https://arxiv.org/abs/1901.05251 %R 10.1016/j.compfluid.2019.05.001 %0 Journal Article %J Computers and Fluids %D 2019 %T A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization %A Michele Girfoglio %A Annalisa Quaini %A Gianluigi Rozza %XWe consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in EFR algorithm. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.

%B Computers and Fluids %V 187 %P 27-45 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85065471890&doi=10.1016%2fj.compfluid.2019.05.001&partnerID=40&md5=c982371b5b5d4b5664a676902aaa60f4 %R 10.1016/j.compfluid.2019.05.001 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2019 %T A Localized Reduced-Order Modeling Approach for PDEs with Bifurcating Solutions %A Martin W. Hess %A Alla, Alessandro %A Annalisa Quaini %A Gianluigi Rozza %A Max Gunzburger %XReduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. In this work, a ROM is developed and tested for the treatment of nonlinear PDEs whose solutions bifurcate as input parameter values change. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does no respect not take advantage of the often large differences in the PDE solutions corresponding to different subregions. Instead, in the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

%B Computer Methods in Applied Mechanics and Engineering %V 351 %P 379-403 %G eng %U https://arxiv.org/abs/1807.08851 %R 10.1016/j.cma.2019.03.050 %0 Journal Article %J Computer Methods in Applied Mechanics and Engineering %D 2019 %T A localized reduced-order modeling approach for PDEs with bifurcating solutions %A Martin W. Hess %A Alla, Alessandro %A Annalisa Quaini %A Gianluigi Rozza %A Max Gunzburger %XReduced-order modeling (ROM) commonly refers to the construction, based on a few solutions (referred to as snapshots) of an expensive discretized partial differential equation (PDE), and the subsequent application of low-dimensional discretizations of partial differential equations (PDEs) that can be used to more efficiently treat problems in control and optimization, uncertainty quantification, and other settings that require multiple approximate PDE solutions. Although ROMs have been successfully used in many settings, ROMs built specifically for the efficient treatment of PDEs having solutions that bifurcate as the values of input parameters change have not received much attention. In such cases, the parameter domain can be subdivided into subregions, each of which corresponds to a different branch of solutions. Popular ROM approaches such as proper orthogonal decomposition (POD), results in a global low-dimensional basis that does not respect the often large differences in the PDE solutions corresponding to different subregions. In this work, we develop and test a new ROM approach specifically aimed at bifurcation problems. In the new method, the k-means algorithm is used to cluster snapshots so that within cluster snapshots are similar to each other and are dissimilar to those in other clusters. This is followed by the construction of local POD bases, one for each cluster. The method also can detect which cluster a new parameter point belongs to, after which the local basis corresponding to that cluster is used to determine a ROM approximation. Numerical experiments show the effectiveness of the method both for problems for which bifurcation cause continuous and discontinuous changes in the solution of the PDE.

%B Computer Methods in Applied Mechanics and Engineering %V 351 %P 379-403 %G eng %U https://www.scopus.com/inward/record.uri?eid=2-s2.0-85064313505&doi=10.1016%2fj.cma.2019.03.050&partnerID=40&md5=8b095034b9e539995facc7ce7bafa9e9 %R 10.1016/j.cma.2019.03.050 %0 Journal Article %J Journal of Computational Physics %D 2017 %T Computational reduction strategies for the detection of steady bifurcations in incompressible fluid-dynamics: Applications to Coanda effect in cardiology %A Giuseppe Pitton %A Annalisa Quaini %A Gianluigi Rozza %K Parametrized Navier–Stokes equations %K Reduced basis method %K Stability of flows %K Symmetry breaking bifurcation %XWe focus on reducing the computational costs associated with the hydrodynamic stability of solutions of the incompressible Navier–Stokes equations for a Newtonian and viscous fluid in contraction–expansion channels. In particular, we are interested in studying steady bifurcations, occurring when non-unique stable solutions appear as physical and/or geometric control parameters are varied. The formulation of the stability problem requires solving an eigenvalue problem for a partial differential operator. An alternative to this approach is the direct simulation of the flow to characterize the asymptotic behavior of the solution. Both approaches can be extremely expensive in terms of computational time. We propose to apply Reduced Order Modeling (ROM) techniques to reduce the demanding computational costs associated with the detection of a type of steady bifurcations in fluid dynamics. The application that motivated the present study is the onset of asymmetries (i.e., symmetry breaking bifurcation) in blood flow through a regurgitant mitral valve, depending on the Reynolds number and the regurgitant mitral valve orifice shape.

In this work we review a reduced basis method for the solution of uncertainty quantification problems. Based on the basic setting of an elliptic partial differential equation with random input, we introduce the key ingredients of the reduced basis method, including proper orthogonal decomposition and greedy algorithms for the construction of the reduced basis functions, a priori and a posteriori error estimates for the reduced basis approximations, as well as its computational advantages and weaknesses in comparison with a stochastic collocation method [I. Babuška, F. Nobile, and R. Tempone, *SIAM Rev.*, 52 (2010), pp. 317--355]. We demonstrate its computational efficiency and accuracy for a benchmark problem with parameters ranging from a few to a few hundred dimensions. Generalizations to more complex models and applications to uncertainty quantification problems in risk prediction, evaluation of statistical moments, Bayesian inversion, and optimal control under uncertainty are also presented to illustrate how to use the reduced basis method in practice. Further challenges, advancements, and research opportunities are outlined.

Read More: http://epubs.siam.org/doi/abs/10.1137/151004550

This monograph addresses the state of the art of reduced order methods for modeling and computational reduction of complex parametrized systems, governed by ordinary and/or partial differential equations, with a special emphasis on real time computing techniques and applications in computational mechanics, bioengineering and computer graphics.

Several topics are covered, including: design, optimization, and control theory in real-time with applications in engineering; data assimilation, geometry registration, and parameter estimation with special attention to real-time computing in biomedical engineering and computational physics; real-time visualization of physics-based simulations in computer science; the treatment of high-dimensional problems in state space, physical space, or parameter space; the interactions between different model reduction and dimensionality reduction approaches; the development of general error estimation frameworks which take into account both model and discretization effects.

This book is primarily addressed to computational scientists interested in computational reduction techniques for large scale differential problems.

%B MS&A %7 1 %I Springer %C Milano %V 9 %P 334 %G eng %6 1 %R 10.1007/978-3-319-02090-7 %0 Journal Article %D 2014 %T A weighted empirical interpolation method: A priori convergence analysis and applications %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method. %I EDP Sciences %G en %U http://urania.sissa.it/xmlui/handle/1963/35021 %1 35253 %2 Mathematics %4 1 %# MAT/05 %$ Approved for entry into archive by Lucio Lubiana (lubiana@sissa.it) on 2015-11-17T10:26:06Z (GMT) No. of bitstreams: 0 %R 10.1051/m2an/2013128 %0 Journal Article %J Communications in Applied and Industrial Mathematics %D 2013 %T Free Form Deformation Techniques Applied to 3D Shape Optimization Problems %A Anwar Koshakji %A Alfio Quarteroni %A Gianluigi Rozza %X The purpose of this work is to analyse and study an efficient parametrization technique for a 3D shape optimization problem. After a brief review of the techniques and approaches already available in literature, we recall the Free Form Deformation parametrization, a technique which proved to be efficient and at the same time versatile, allowing to manage complex shapes even with few parameters. We tested and studied the FFD technique by establishing a path, from the geometry definition, to the method implementation, and finally to the simulation and to the optimization of the shape. In particular, we have studied a bulb and a rudder of a race sailing boat as model applications, where we have tested a complete procedure from Computer-Aided-Design to build the geometrical model to discretization and mesh generation. %B Communications in Applied and Industrial Mathematics %G eng %R 10.1685/journal.caim.452 %0 Journal Article %J SIAM Journal on Scientific Computing %D 2013 %T Reduced basis method for parametrized elliptic optimal control problems %A Federico Negri %A Gianluigi Rozza %A Andrea Manzoni %A Alfio Quarteroni %X We propose a suitable model reduction paradigm-the certified reduced basis method (RB)-for the rapid and reliable solution of parametrized optimal control problems governed by partial differential equations. In particular, we develop the methodology for parametrized quadratic optimization problems with elliptic equations as a constraint and infinite-dimensional control variable. First, we recast the optimal control problem in the framework of saddle-point problems in order to take advantage of the already developed RB theory for Stokes-type problems. Then, the usual ingredients of the RB methodology are called into play: a Galerkin projection onto a low-dimensional space of basis functions properly selected by an adaptive procedure; an affine parametric dependence enabling one to perform competitive offline-online splitting in the computational procedure; and an efficient and rigorous a posteriori error estimate on the state, control, and adjoint variables as well as on the cost functional. Finally, we address some numerical tests that confirm our theoretical results and show the efficiency of the proposed technique. %B SIAM Journal on Scientific Computing %V 35 %P A2316–A2340 %G eng %R 10.1137/120894737 %0 Report %D 2013 %T A Reduced Computational and Geometrical Framework for Inverse Problems in Haemodynamics %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %I SISSA %G en %1 6571 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-30T14:47:07Z No. of bitstreams: 1 LMQR_inverse_problems_Haemo.pdf: 2913600 bytes, checksum: 9bff594072e6c7b4a664bdade361a1a9 (MD5) %0 Report %D 2013 %T A reduced-order strategy for solving inverse Bayesian identification problems in physiological flows %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %I SISSA %G en %1 6555 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-04-12T10:16:42Z\nNo. of bitstreams: 1\nHanoi_RQML_2012_REVISED.pdf: 1360222 bytes, checksum: 4cb9ffaaf9fd67e3eacf4d9c060d0940 (MD5) %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T Stochastic optimal robin boundary control problems of advection-dominated elliptic equations %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we deal with a stochastic optimal Robin boundary control problem constrained by an advection-diffusion-reaction elliptic equation with advection-dominated term. We assume that the uncertainty comes from the advection field and consider a stochastic Robin boundary condition as control function. A stochastic saddle point system is formulated and proved to be equivalent to the first order optimality system for the optimal control problem, based on which we provide the existence and uniqueness of the optimal solution as well as some results on stochastic regularity with respect to the random variables. Stabilized finite element approximations in physical space and collocation approximations in stochastic space are applied to discretize the optimality system. A global error estimate in the product of physical space and stochastic space for the numerical approximation is derived. Illustrative numerical experiments are provided. %B SIAM Journal on Numerical Analysis %V 51 %P 2700–2722 %G eng %R 10.1137/120884158 %0 Journal Article %J SIAM Journal on Numerical Analysis %D 2013 %T A weighted reduced basis method for elliptic partial differential equations with random input data %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %X In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems. %B SIAM Journal on Numerical Analysis %V 51 %P 3163–3185 %G eng %R 10.1137/130905253 %0 Journal Article %J Mathematical Modelling and Numerical Analysis, in press, 2012-13 %D 2012 %T Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %K shape optimization %X We review the optimal design of an arterial bypass graft following either a (i) boundary optimal control approach, or a (ii) shape optimization formulation. The main focus is quantifying and treating the uncertainty in the residual flow when the hosting artery is not completely occluded,\\r\\nfor which the worst-case in terms of recirculation e ffects is inferred to correspond to a strong ori fice flow through near-complete occlusion. A worst-case optimal control approach is applied to the steady\\r\\nNavier-Stokes equations in 2D to identify an anastomosis angle and a cu ed shape that are robust with respect to a possible range of residual \\r\\nflows. We also consider a reduced order modelling framework\\r\\nbased on reduced basis methods in order to make the robust design problem computationally feasible. The results obtained in 2D are compared with simulations in a 3D geometry but without model\\r\\nreduction or the robust framework. %B Mathematical Modelling and Numerical Analysis, in press, 2012-13 %I Cambridge University Press %G en %U http://hdl.handle.net/1963/6337 %1 6267 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T15:38:41Z\\nNo. of bitstreams: 1\\nLMQR_M2AN_Special_SISSAreport.pdf: 5702019 bytes, checksum: 037e51bde713582eff1ee9766b1b4559 (MD5) %0 Book Section %B Springer, Indam Series, Vol. 4, 2012 %D 2012 %T Generalized reduced basis methods and n-width estimates for the approximation of the solution manifold of parametric PDEs %A Toni Lassila %A Andrea Manzoni %A Alfio Quarteroni %A Gianluigi Rozza %K solution manifold %X The set of solutions of a parameter-dependent linear partial di fferential equation with smooth coe fficients typically forms a compact manifold in a Hilbert space. In this paper we review the generalized reduced basis method as a fast computational tool for the uniform approximation of the solution manifold. We focus on operators showing an affi ne parametric dependence, expressed as a linear combination of parameter-independent operators through some smooth, parameter-dependent scalar functions. In the case that the parameter-dependent operator has a dominant term in its affi ne expansion, one can prove the existence of exponentially convergent uniform approximation spaces for the entire solution manifold. These spaces can be constructed without any assumptions on the parametric regularity of the manifold \\r\\nonly spatial regularity of the solutions is required. The exponential convergence rate is then inherited by the generalized reduced basis method. We provide a numerical example related to parametrized elliptic\\r\\nequations con rming the predicted convergence rates. %B Springer, Indam Series, Vol. 4, 2012 %I Springer %G en %U http://hdl.handle.net/1963/6340 %1 6270 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2012-12-13T17:47:45Z\\nNo. of bitstreams: 1\\nqlmr-bumi_FINAL_SISSAreport.pdf: 377397 bytes, checksum: ecaf5713afa6a6f68992f6331631aff4 (MD5) %0 Journal Article %J International Journal Numerical Methods Biomedical Engineering %D 2012 %T Simulation-based uncertainty quantification of human arterial network hemodynamics %A Peng Chen %A Alfio Quarteroni %A Gianluigi Rozza %K uncertainty quantification, mathematical modelling of the cardiovascular system, fluid-structure interaction %X This work aims at identifying and quantifying uncertainties from various sources in human cardiovascular\r\nsystem based on stochastic simulation of a one dimensional arterial network. A general analysis of\r\ndifferent uncertainties and probability characterization with log-normal distribution of these uncertainties\r\nis introduced. Deriving from a deterministic one dimensional fluid structure interaction model, we establish\r\nthe stochastic model as a coupled hyperbolic system incorporated with parametric uncertainties to describe\r\nthe blood flow and pressure wave propagation in the arterial network. By applying a stochastic collocation\r\nmethod with sparse grid technique, we study systemically the statistics and sensitivity of the solution with\r\nrespect to many different uncertainties in a relatively complete arterial network with potential physiological\r\nand pathological implications for the first time. %B International Journal Numerical Methods Biomedical Engineering %I Wiley %G en %1 6467 %2 Mathematics %4 1 %# MAT/08 ANALISI NUMERICA %$ Submitted by Gianluigi Rozza (grozza@sissa.it) on 2013-03-07T12:33:02Z\nNo. of bitstreams: 1\nreport.pdf: 1605405 bytes, checksum: bb12cf074ce32a80567a0cde0c0861db (MD5) %0 Journal Article %J Int. J. Numer. Meth. Fluids %D 2004 %T Calculation of impulsively started incompressible viscous flows %A Marra, Andrea %A Andrea Mola %A Quartapelle, Luigi %A Riviello, Luca %B Int. J. Numer. Meth. Fluids %V 46 %P 877–902 %G eng